Chapter 7.7, Problem 20P

Interpretation Introduction

Interpretation:

The Coefficient of performance

Concept Introduction:

Write the expression for the difference in entropy using residuals.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1=\left({\underset{\_}{S}}_2-{\underset{\_}{S}}_2^{ig}\right)+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}\right)$

Here, molar entropy at state 1 (liquid) and 2 (vapor) are ${\underset{\_}{S}}_1$ and ${\underset{\_}{S}}_2$, molar entropy at state 1 and 2 at inert gas are ${\underset{\_}{S}}_1^{ig}$ and ${\underset{\_}{S}}_2^{ig}$ respectively.

Write the ideal gas entropy change.

${\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}=C_P^{*}\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{P_2}{P_1}\right)$

Here, temperature and pressure at state 1 and 2 is $T_1,T_2,P_1,\text{and}{P}_2$, gas constant is R, and heat capacity at constant pressure for an ideal gas is $C_P^{*}$.

Write the van der Waals EOS.

$P_2=\frac{R{T}_2}{{\underset{\_}{V}}_2-b}-\frac{a}{{\underset{\_}{V}}_2^2}$

Here, van der Waals parameter are a and b, molar volume at state 2 is ${\underset{\_}{V}}_2$, and gas constant is R.

Write the expression for the residual entropy obtained from the van der Waals EOS.

$\begin{array}{c}{\underset{\_}{S}}_2-{\underset{\_}{S}}_2{}^{ig}=R\ln \left(Z\right)+R\ln \left(\frac{{\underset{\_}{V}}_2-b}{{\underset{\_}{V}}_2}\right)\\ =R\ln \left(\frac{P_2{\underset{\_}{V}}_2}{R{T}_2}\right)+R\ln \left(\frac{{\underset{\_}{V}}_2-b}{{\underset{\_}{V}}_2}\right)\end{array}$

Here, compressibility factor is Z.

Write the residual molar enthalpy according to the van der Waals EOS.

${\underset{\_}{H}}^R=RT\left(Z-1\right)+{\displaystyle \underset{T=T,\underset{\_}{V}=\infty }{\overset{T=T,\underset{\_}{V}=\underset{\_}{V}}{\int }}\left[T{\left(\frac{\partial P}{\partial T}\right)}_{\underset{\_}{V}}-P\right]}d\underset{\_}{V}$

Here, change in pressure with respect to change in temperature at constant molar volume is ${\left(\frac{\partial P}{\partial T}\right)}_{\underset{\_}{V}}$.

Write the change in molar enthalpy using residual properties across the compressor.

$\frac{{\dot{W}}_{S,rev}}{\dot{n}}=\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)$

Here, molar enthalpy at state 1 (liquid) and 2 (vapor) is ${\underset{\_}{H}}_1$ and ${\underset{\_}{H}}_2$, molar enthalpy at state 1 and 2 at inert gas is ${\underset{\_}{H}}_1^{ig}$ and ${\underset{\_}{H}}_2^{ig}$ respectively

Write the efficiency of the compressor.

$\begin{array}{l}\eta =\frac{\frac{{\dot{W}}_{S,rev}}{\dot{n}}}{\frac{{\dot{W}}_{S,act}}{\dot{n}}}\\ \frac{{\dot{W}}_{S,act}}{\dot{n}}=\frac{\frac{{\dot{W}}_{S,rev}}{\dot{n}}}{\eta }\end{array}$   

Here, actual rate of work done on the shaft is ${\dot{W}}_{S,act}$, molar flow rate is $\dot{n}$, and efficiency is $\eta$.

Write the energy balance on the entire refrigeration cycle.

$0=\frac{{\dot{Q}}_H}{\dot{n}}+\frac{{\dot{Q}}_C}{\dot{n}}+\frac{{\dot{W}}_{S,act}}{\dot{n}}$

Write the coefficient of performance.

$\text{COP}=\frac{\frac{{\dot{Q}}_C}{\dot{n}}}{\frac{{\dot{W}}_{S,act}}{\dot{n}}}$